The dossier includes a number of salacious claims about the Trump campaign's ties to Russian officials, including allegations about Mr Cohen that have not been verified.
Both Mr Cohen's lawsuits claim the dossier contained "false and defamatory" allegations about him that resulted in "harm to his personal and professional reputation, current business interests, and the impairment of business opportunities," ABC News reported.
"Enough is enough of the #fake #RussianDossier," Mr Cohen tweeted on Tuesday. "Just filed a defamation action against @BuzzFeedNews for publishing the lie filled document on @POTUS @realDonaldTrump and me!"
BuzzFeed responded with a Twitter post saying the dossier continues to be the subject of investigations by Congress and intelligence agencies, was shown to two US presidents and described in detail by news outlets worldwide.
"Its interest to the public is obvious. This is not the first time Trump's lawyer has attacked the free press, and we look forward to defending our First Amendment rights in court," BuzzFeed said.
The lawsuits were not immediately available on online court resources.
It comes amid reports that Glenn Simpson, who leads Fusion GPS, testified in August that the FBI received intelligence from a "voluntary source" in Mr Trump's circle in 2016.
That was before Mr Steele gave the FBI his own dossier of allegations about collusion. Fusion GPS then hired Mr Steele to investigate Mr Trump's connections with Russia.
Some Republicans have been critical of Special Counsel Robert Mueller's probe into possible collusion between the Trump campaign and Russia, claiming the initial FBI probe was triggered by Mr Steele's dossier.
They have raised questions about whether the FBI may have relied on the Steele document to improperly obtain surveillance warrants to spy on Mr Trump's campaign associates.
But Mr Simpson's testimony before the Senate Judiciary Committee in August contradicted those claims.
"They had other intelligence about this matter from an internal Trump campaign source and ... they believed Chris' information might be credible," Mr Simpson testified. "I think it was a voluntary source, someone who was concerned about the same concerns we had."
Mr Simpson later said he was not sure if the FBI informant was from the Mr Trump's election campaign or the Trump Organisation, the president's real estate firm, which is now run by his two older sons.
Dianne Feinstein, the ranking Democrat on the Senate committee, released the testimony on Tuesday without Republican approval, in an apparent bid to neutralise Republican criticism of the Russia investigation.
Mr Trump has slammed the dossier as "bogus" and denies his campaign colluded with Russia.
Mr Trump and some Republicans have attacked both Mr Simpson's firm and Mr Steele in recent months and also questioned the impartiality of the FBI and the Justice Department in an apparent effort to discredit Mr Mueller's probe.
Mr Mueller's team has produced two indictments and two guilty pleas by former aides to Mr Trump's campaign.
The original article was published in the Independent.
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